Daily Prep 6A -- MTH 201-04

--- tags: mth201, dailyprep --- # Daily Prep 6A -- MTH 201-04 ## Overview This module introduces the final and perhaps most important derivative rule we will learn in the course: **The Chain Rule**. The Chain Rule is used when taking derivatives of *composite* functions, which you'll recall are functions that are made up of two or more functions "chained" together by plugging one into the other, like $\sqrt{x^2 + 1}$ or $\cos(e^x)$. The commonality of composites in real applications makes the Chain Rule of primary importance as we move ahead. ## Learning objectives **Basic Learning Objectives:** *Before* our class meeting, use the Resources listed below to learn all of the following. You should be reasonably fluent with all of these tasks prior to our meeting; we will field questions on these, but they will not be retaught. + (Review) Determine the fundamental algebraic structure of a function. + (Review) If the fundamental algebraic structure of a function is a *composite*, determine the inner and outer functions in the composite. + State the Chain Rule and apply it to a simple composite function. **Advanced Learning Objectives:** *During and after* our class meeting, we will work on learning the following. Fluency with these is not required prior to class. + Use the Chain Rule to differentiate a composite of two basic functions. + Differentiate a function whose derivative involves a mixture of rules (Product, Quotient, Chain, etc.) ## Resources for learning **Because of the amount of video examples this time, I recommend doing the video first and then read later.** **Reading:** Read through [Section 2.5](https://activecalculus.org/single/sec-2-5-chain.html) of the *Active Calculus* textbook. Work through the examples and all interactive exercises found at the end of the section. **Video:** There's quite a bit of video to watch this time -- if you watch all of these below, the complete running time is 51 minutes, 20 seconds. (You are *not* required to watch all of them --- you can pick and choose, skip parts, play at 2x speed, etc. but these are here for you to learn, so don't skip just to avoid working.) **You should approach these videos as if they were an entire, single class meeting of nothing but lecture and examples**. First, review the two videos on composite functions and fundamental algebraic structure that were made for the Precalculus Review assignment during startup week. You can find those on Blackboard > *START HERE* > *Precalculus Review Assignment* > *Tutorial Videos* and then these are videos 7 and 8. They've also been re-posted in the *Module 6* folder just below this Daily Prep. Or click the links: - (Video 7) Composites: https://gvsu.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=8742a5e1-3206-4e92-b3ac-ac240134cd86 - (Video 8) Fundamental structure: https://gvsu.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=672d8d71-903a-4965-8109-ac240134e943 Then, watch the following from the GVSUMath Calculus playlist: - Screencast 2.5.1: Quick review - The Chain Rule (2:21) https://www.youtube.com/watch?v=HxVn6kRD5NM&list=PL9bIjQJDwfGuXQHuS5Jkmum_CFILoCZX-&index=39 - Screencast 2.5.2: Example of the Chain Rule - Polynomials (4:33) https://www.youtube.com/watch?v=QDc1UmLWhug&list=PL9bIjQJDwfGuXQHuS5Jkmum_CFILoCZX-&index=40 - Screencast 2.5.3: Example of the Chain Rule - Radicals (7:20) https://www.youtube.com/watch?v=ysp96e3Z-nw&list=PL9bIjQJDwfGuXQHuS5Jkmum_CFILoCZX-&index=41 - Screencast 2.5.4: Chain Rule examples - Trigonometric functions (5:15) https://www.youtube.com/watch?v=4y39u0DmrPY&list=PL9bIjQJDwfGuXQHuS5Jkmum_CFILoCZX-&index=42 - Screencast 2.5.5: Chain Rule examples - Exponential functions (6:01) https://www.youtube.com/watch?v=zexX6t_zbCg&list=PL9bIjQJDwfGuXQHuS5Jkmum_CFILoCZX-&index=43 - Screencast 2.5.6: Chain Rule examples - Mixing rules (10:30) https://www.youtube.com/watch?v=1B06Pk3W6Pc&list=PL9bIjQJDwfGuXQHuS5Jkmum_CFILoCZX-&index=44 - Screencast 2.5.7: Chain Rule examples - Graphs only (5:38) https://www.youtube.com/watch?v=pwm50foAx6A&list=PL9bIjQJDwfGuXQHuS5Jkmum_CFILoCZX-&index=45 You are free to search for and use other resources in addition to, or instead of the above, as long as you can work the exercises below. ## Exercises The exercises for this Daily Prep are found on Desmos. Go to student.desmos.com and look for "MTH 201 Daily Prep 6A". ## Submission and grading To submit your work, simply do the work on Desmos and your work is saved every time you do something on screen. A **Pass** mark is given if the Daily Prep is turned in before its deadline and if each item on the Daily Prep has a response that represents a good faith effort to be right. **Mistakes are not penalized**. A **No Pass** is given if an item is left blank (even accidentally), has an answer but it shows insufficient effort (including responses like "I don't know"), or if the Daily Prep is late.